Olympiad problems

2010 IMAC Arhimede, 3

Tags: trigonometry , geometry

Let $ABC$ be a triangle and let $D\in (BC)$ be the foot of the $A$- altitude. The circle $w$ with the diameter $[AD]$ meet again the lines $AB$ , $AC$ in the points $K\in (AB)$ , $L\in (AC)$ respectively. Denote the meetpoint $M$ of the tangents to the circle $w$ in the points $K$ , $L$ . Prove that the ray $[AM$ is the $A$-median in $\triangle ABC$ ([b][u]Serbia[/u][/b]).

2014 Contests, 2

Tags: number theory

Solve the following equation in $\mathbb{Z}$: \[3^{2a + 1}b^2 + 1 = 2^c\]

2012 USA TSTST, 6

Tags: inequalities , algebra , polynomial , vieta , quadratic

Positive real numbers $x, y, z$ satisfy $xyz+xy+yz+zx = x+y+z+1$. Prove that \[ \frac{1}{3} \left( \sqrt{\frac{1+x^2}{1+x}} + \sqrt{\frac{1+y^2}{1+y}} + \sqrt{\frac{1+z^2}{1+z}} \right) \le \left( \frac{x+y+z}{3} \right)^{5/8} . \]

2005 Romania National Olympiad, 4

Tags: inequalities , algebra

a) Prove that for all positive reals $u,v,x,y$ the following inequality takes place: \[ \frac ux + \frac vy \geq \frac {4(uy+vx)}{(x+y)^2} . \] b) Let $a,b,c,d>0$. Prove that \[ \frac a{b+2c+d} + \frac b{c+2d+a} + \frac c{d+2a+b} + \frac d{a+2b+c} \geq 1.\] [i]Traian Tămâian[/i]

2013 Greece Team Selection Test, 1

Tags: number theory

Find all pairs of non-negative integers $(m,n)$ satisfying $\frac{n(n+2)}{4}=m^4+m^2-m+1$

2016 Purple Comet Problems, 16

Tags: geometry , rectangle

The figure below shows a barn in the shape of two congruent pentagonal prisms that intersect at right angles and have a common center. The ends of the prisms are made of a 12 foot by 7 foot rectangle surmounted by an isosceles triangle with sides 10 feet, 10 feet, and 12 feet. Each prism is 30 feet long. Find the volume of the barn in cubic feet. [center][img]https://snag.gy/Ox9CUp.jpg[/img][/center]

2016 All-Russian Olympiad, 3

Tags: number theory , greatest common divisor , divisor

Alexander has chosen a natural number $N>1$ and has written down in a line,and in increasing order,all his positive divisors $d_1<d_2<\ldots <d_s$ (where $d_1=1$ and $d_s=N$).For each pair of neighbouring numbers,he has found their greater common divisor.The sum of all these $s-1$ numbers (the greatest common divisors) is equal to $N-2$.Find all possible values of $N$.

JOM 2015 Shortlist, G5

Tags: geometry

Let $ ABCD $ be a convex quadrilateral. Let angle bisectors of $ \angle B $ and $ \angle C $ intersect at $ E $. Let $ AB $ intersect $ CD $ at $ F $. Prove that if $ AB+CD=BC $, then $A,D,E,F$ is cyclic.

1956 AMC 12/AHSME, 30

Tags: geometry

If the altitude of an equilateral triangle is $ \sqrt {6}$, then the area is: $ \textbf{(A)}\ 2\sqrt {2} \qquad\textbf{(B)}\ 2\sqrt {3} \qquad\textbf{(C)}\ 3\sqrt {3} \qquad\textbf{(D)}\ 6\sqrt {2} \qquad\textbf{(E)}\ 12$

1994 India Regional Mathematical Olympiad, 2

Tags: geometry , incenter , area of a triangle , heron's formula

In a triangle $ABC$, the incircle touches the sides $BC, CA, AB$ at $D, E, F$ respectively. If the radius if the incircle is $4$ units and if $BD, CE , AF$ are consecutive integers, find the sides of the triangle $ABC$.

2020 Taiwan TST Round 3, 2

Tags: number theory , floor function

Let $H = \{ \lfloor i\sqrt{2}\rfloor : i \in \mathbb Z_{>0}\} = \{1,2,4,5,7,\dots \}$ and let $n$ be a positive integer. Prove that there exists a constant $C$ such that, if $A\subseteq \{1,2,\dots, n\}$ satisfies $|A| \ge C\sqrt{n}$, then there exist $a,b\in A$ such that $a-b\in H$. (Here $\mathbb Z_{>0}$ is the set of positive integers, and $\lfloor z\rfloor$ denotes the greatest integer less than or equal to $z$.)

2017 Online Math Open Problems, 30

Tags:

Let $p = 2017$ be a prime. Given a positive integer $n$, let $T$ be the set of all $n\times n$ matrices with entries in $\mathbb{Z}/p\mathbb{Z}$. A function $f:T\rightarrow \mathbb{Z}/p\mathbb{Z}$ is called an $n$-[i]determinant[/i] if for every pair $1\le i, j\le n$ with $i\not= j$, \[f(A) = f(A'),\] where $A'$ is the matrix obtained by adding the $j$th row to the $i$th row. Let $a_n$ be the number of $n$-determinants. Over all $n\ge 1$, how many distinct remainders of $a_n$ are possible when divided by $\dfrac{(p^p - 1)(p^{p - 1} - 1)}{p - 1}$? [i]Proposed by Ashwin Sah[/i]

1954 AMC 12/AHSME, 39

Tags: geometry , similar triangles

The locus of the midpoint of a line segment that is drawn from a given external point $ P$ to a given circle with center $ O$ and radius $ r$, is: $ \textbf{(A)}\ \text{a straight line perpendicular to }\overline{PO} \\ \textbf{(B)}\ \text{a straight line parallel to } \overline{PO} \\ \textbf{(C)}\ \text{a circle with center }P\text{ and radius }r \\ \textbf{(D)}\ \text{a circle with center at the midpoint of }\overline{PO}\text{ and radius }2r \\ \textbf{(E)}\ \text{a circle with center at the midpoint }\overline{PO}\text{ and radius }\frac{1}{2}r$

1977 AMC 12/AHSME, 6

Tags:

If $x, y$ and $2x + \frac{y}{2}$ are not zero, then \[ \left( 2x + \frac{y}{2} \right)\left[(2x)^{-1} + \left( \frac{y}{2} \right)^{-1} \right] \] equals $\textbf{(A) }1\qquad\textbf{(B) }xy^{-1}\qquad\textbf{(C) }x^{-1}y\qquad\textbf{(D) }(xy)^{-1}\qquad \textbf{(E) }\text{none of these}$

2010 Federal Competition For Advanced Students, P2, 6

Tags: circumcircle , hexagon , combinatorial geometry , geometry

A diagonal of a convex hexagon is called [i]long[/i] if it decomposes the hexagon into two quadrangles. Each pair of [i]long[/i] diagonals decomposes the hexagon into two triangles and two quadrangles. Given is a hexagon with the property, that for each decomposition by two [i]long[/i] diagonals the resulting triangles are both isosceles with the side of the hexagon as base. Show that the hexagon has a circumcircle.

2006 AMC 10, 14

Tags:

Let $ a$ and $ b$ be the roots of the equation $ x^2 \minus{} mx \plus{} 2 \equal{} 0$. Suppose that $ a \plus{} (1/b)$ and $ b \plus{} (1/a)$ are the roots of the equation $ x^2 \minus{} px \plus{} q \equal{} 0$. What is $ q$? $ \textbf{(A) } \frac 52 \qquad \textbf{(B) } \frac 72 \qquad \textbf{(C) } 4 \qquad \textbf{(D) } \frac 92 \qquad \textbf{(E) } 8$

2019 AMC 10, 4

Tags: counting

A box contains $28$ red balls, $20$ green balls, $19$ yellow balls, $13$ blue balls, $11$ white balls, and $9$ black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least $15$ balls of a single color will be drawn$?$ $\textbf{(A) } 75 \qquad\textbf{(B) } 76 \qquad\textbf{(C) } 79 \qquad\textbf{(D) } 84 \qquad\textbf{(E) } 91$

2011 Mediterranean Mathematics Olympiad, 2

Tags: ratio , inequalities , function , geometric sequence , combinatorics

Let $A$ be a finite set of positive reals, let $B = \{x/y\mid x,y\in A\}$ and let $C = \{xy\mid x,y\in A\}$. Show that $|A|\cdot|B|\le|C|^2$. [i](Proposed by Gerhard Woeginger, Austria)[/i]

1985 Traian Lălescu, 1.4

Tags: geometry , parallelogram , pure geometry

Let $ ABCD $ be a convex quadrilateral, and $ P $ be a point that isn't found on any of the lines formed by the sides of the quadrilateral. Prove that the centers of mass of the triangles $ PAB, PBC, PCD $ and $ PDA, $ form a parallelogram, and calculate the legths of its sides in terms of its diagonals.

2022 3rd Memorial "Aleksandar Blazhevski-Cane", P1

Tags: geometry

Let $ABC$ be an acute triangle with altitude $AD$ ($D \in BC$). The line through $C$ parallel to $AB$ meets the perpendicular bisector of $AD$ at $G$. Show that $AC = BC$ if and only if $\angle AGC = 90^{\circ}$.

1991 Irish Math Olympiad, 3

Tags: function , algebra

Three operations $f,g$ and $h$ are defined on subsets of the natural numbers $\mathbb{N}$ as follows: $f(n)=10n$, if $n$ is a positive integer; $g(n)=10n+4$, if $n$ is a positive integer; $h(n)=\frac{n}{2}$, if $n$ is an [i]even[/i] positive integer. Prove that, starting from $4$, every natural number can be constructed by performing a finite number of operations $f$, $g$ and $h$ in some order. $[$For example: $35=h(f(h(g(h(h(4)))))).]$

KoMaL A Problems 2017/2018, A. 721

Tags: inequalities , algebra

Let $n\ge 2$ be a positive integer, and suppose $a_1,a_2,\cdots ,a_n$ are positive real numbers whose sum is $1$ and whose squares add up to $S$. Prove that if $b_i=\tfrac{a^2_i}{S} \;(i=1,\cdots ,n)$, then for every $r>0$, we have $$\sum_{i=1}^n \frac{a_i}{{(1-a_i)}^r}\le \sum_{i=1}^n \frac{b_i}{{(1-b_i)}^r}.$$

2024-25 IOQM India, 28

Tags:

Find the largest positive integer $n <30$ such that $\frac{1}{2}(n^8 + 3n^4 -4)$ is not divisible by the square of any prime number.

2011 HMNT, 9

Tags: geometry

Let $ABC$ be a triangle with $AB = 13$, $BC = 14$, and $CA = 15$. Let $D$ be the foot of the altitude from $A$ to $BC$. The inscribed circles of triangles $ABD$ and $ACD$ are tangent to $AD$ at $P$ and $Q$, respectively, and are tangent to $BC$ at $X$ and $Y$ , respectively. Let $PX$ and $QY$ meet at $Z$. Determine the area of triangle $XY Z$.

2015 Harvard-MIT Mathematics Tournament, 5

Tags: algebra , inequalities , maximization

Let $a,b,c$ be positive real numbers such that $a+b+c=10$ and $ab+bc+ca=25$. Let $m=\min\{ab,bc,ca\}$. Find the largest possible value of $m$.